4. Games & Strategies
Games and Strategies
Simultaneous Games
Sequential Games
Repeated Games
}
}
static
omamicy
}
2
Simultaneous Game
Ex: Prisoner’s Dilemma
3
L
R
T
5, 5
3, 6
B
6, 3
4, 4
Prisoner’s Dilemma
4
Cooperate
Defect
Cooperate
Win, Win
Lose much,
Win much
Defect
Win much,
Lose much
Lose, Lose
Three Components of Game
} Players
} Strategies
} Payoff
5
Illustration of Games
} Normal
form (strategic form)
} Extensive form (game tree)
다
6
Dominant Strategy
Dominant strategy: A strategy that is strictly better
than any other strategy regardless of the other
<
players’ strategy choices.
xefect ,
Dominated strategy: the strategy whose payoff is
inferior to that of another strategy, regardless of
what the other player does
}
}
ccooperatey
7
Assumptions
Rationality
Common Knowledge
}
}
}
}
}
}
8
Players are rational.
Each player believes that the other player is rational.
Each player believes that the other player believes that
the first player is rational.
And so on…
Iterated Elimination of Dominated
Strategies
9
L
C
R
T
1, 1
2, 0
1, 1
M
0, 0
0, 1
0, 0
B
2, 1
1, 0
2, 2
Iterated Elimination of Dominated
Strategies
10
L
C
R
T
1, 1
2, 0
1, 1
M
0, 0
0, 1
0, 0
B
2, 1
1, 0
2, 2
Iterated Elimination of Dominated
Strategies
11
L
C
R
T
1, 1
2, 0
1, 1
M
0, 0
0, 1
0, 0
B
2, 1
1, 0
2, 2
Iterated Elimination of Dominated
Strategies
12
L
C
R
T
1, 1
2, 0
1, 1
M
0, 0
0, 1
0, 0
B
2, 1
1, 0
2, 2
Iterated Elimination of Dominated
Strategies
13
L
C
R
T
1, 1
2, 0
1, 1
M
0, 0
0, 1
0, 0
B
2, 1
1, 0
2, 2
Dubious Application of Dominated
Strategies
14
L
R
T
1, 0
1, 1
B
-100, 0
2, 1
Dubious Application of Dominated
Strategies
15
L
R
T
1, 0
1, 1
B
-100, 0
2, 1
Dubious Application of Dominated
Strategies
16
L
R
T
1, 0
1, 1
B
-100, 0
2, 1
Neither Dominant nor Dominated
17
L
C
R
T
2, 1
1, 3
0, 2
M
1, 2
5, 1
1, 1
B
0, 1
0, 0
2, 2
Nash Equilibrium
Players choose an optimal strategy given
their conjectures of what the other players
do. Cexpe cfation)
2.
Such conjectures are consistent with the
other players’ strategy choices.
à Nash equilibrium: if no player can unilaterally
change its strategy in a way that improve its
payoff.
1.
18
Nash Equilibrium
}
19
A configuration of strategies, one for each
player, such that each player’s strategy is best
for her given that all the other players are
playing their equilibrium strategies
John Forbes Nash (1928 ~ 2015)
Nobel Prize (1994)
Mathematician @ Princeton U.
20
A Beautiful Mind (2002)
21
Russell Crowe (1964 ~)
S=
Nash Equilibrium
TiSS
=
(
*
1
SL
C
,
M , B3
,
RS
X
1비
*
2
( s , s ) is a Nash equilibrium iff
u1 ( s , s ) ³ u1 ( s1 , s ) for any s1 Î S1
*
1
*
2
*
1
*
2
*
2
[ u (s , s ) ³ u (s , s ) for any s
.
2
22
2
*
1
2
2
Î S2
.
*
Nash Equilibrium
( s1* , s2* ) is a Nash equilibrium iff
u1 ( s1* , s2e ) ³ u1 ( s1 , s2e ) for any s1 Î S1
u 2 ( s1e , s2* ) ³ u2 ( s1e , s2 ) for any s2 Î S 2
s2e = s2* , s1e = s1*
23
Dominant Strategy Equilibrium
*
1
*
2
( s , s ) is a D.S.E. iff
u1 ( s , s2 ) ³ u1 ( s1 , s2 ) for any s1 Î S1 , s2 Î S 2
*
1
u2 ( s1 , s2* ) ³ u2 ( s1 , s2 ) for any s1 Î S1 , s2 Î S 2
24
Coordination Game
- Multiple Nash Equilibria
25
L
R
T
1, 1
0, 0
B
0, 0
1, 1
Battle of the Sexes
- Multiple Nash Equilibria
X
26
L
(Opera)
R
(Soccer)
T
(Opera)
1, 2
0, 0
B
(Soccer)
0, 0
2, 1
Sequential Game
Ex: Entry Game
}
A: Potential entrant
}
}
Enter, Do not enter
B: Incumbent
}
Retaliate, Do not retaliate
redatory
1
27
priciag
Entry Game
-10, -10
B
r
e
Not r
10, 20
A
Not e
0, 50
28
by cell
Entry Game in Normal Form
inspencfion
cell
29
-
-
r
Not r
e
-10, -10
10, 20
Not e
0, 50
0, 50
Nash Equilibrium
}
Two N.E.
}
}
30
(e, Not r)
(Not e, r)
Rollback Equilibrium (backward induction)
-10, -10
B
r
e
Not r
10, 20
A
Not e
0, 50
31
Rollback Equilibrium (backward induction)
-10, -10
B
r
e
Not r
10, 20
A
Not e
0, 50
32
Incredible Threat
}
Consider a N.E. (Not e, r)
}
r: incredible threat
}
Once player A enters the market, “Not r” is the optimal
strategy in that situation. Playing “r” is not good even for
player B.
…
33
Subgame
}
A subgame in an extensive-form game
3
Begins in at a decision node n that is a singleton
information set.
} Includes all the decision and terminal nodes
following n in the game (but no nodes that do not
follow n).
} Does not cut any information set (i.e., if a decision
"
node n’ follows n in the game tree, then all other
nodes in the information set containing n’ must also
follow n, and so must be included in the subgame).
}
fn
1. 2 x 2 simultaneous game
2. Signaling game
34
setfbset DOIM
A
=
y
Kiz
sis
3
S"용
…
Subgame_simultaneous game
5, 5
B
L
U
R
3, 6
A
6, 3
D
T
R
35
4, 4
SPNE
}
Definition
}
36
A Nash equilibrium is subgame-perfect if the players’
strategies constitute a Nash equilibrium in every
subgame.
Rollback Equilibrium (backward induction)
-10, -10
B
r
e
Not r
10, 20
A
Not e
0, 50
32
Value of Commitment (Figure 7_p.57)
}
Commitment
}
Player B commits to choose r if player A chooses e.
}
}
}
}
}
If player B choose Not r, penalty = 40.
This commitment is credible.(-20 < -10)
Payoff (commitment) = 50
Payoff (No commitment) = 20
Value of commitment = 30
37
38
}
}
A commitment device is a way to lock yourself
into following a plan of action that you might not
want to do but you know is good for you.
Chinese military general Han Xin purportedly
created a commitment device for his soldiers: he
placed them with their backs to a river to make
sure they would fight.
39
Examples of Commitment
}
}
}
Large advertising
Purchase of inputs in advance
Take large orders in advance
40
cf. Sequential Version of P.D.
}
Game Tree?
41
Action
F
Sequenfial
C
Gome
정
-
sfrafegy
5
.
T
R
~
A
B
-
3
0
6
.
.
3
R
~> *
Normal Form
(S,
A
, SA)
2
LL
LR
RL
RR
T
5,5
5,5
3,6
3,6
B
6,3
4,4
6,3
4,4
1
42
에
Repeated Games
}
}
One-shot game
Repeated game
}
A one-shot game (or stage game) is repeated a
number of times
}
}
43
Finitely repeated game
Infinitely repeated game
Stage Game
Actious
=
44
Strategies
L
C
R
T
5, 5
3, 6
0, 0
M
6, 3
4, 4
0, 0
B
0, 0
0, 0
1, 1
Strategy in Twice Repeated Games
}
Strategy (vs. action)
}
}
A player’s complete contingent plan of action for all
possible occurrences in the game
Player 1(2)’s strategy
}
}
What to choose in period 1: 3
What to choose in period 2 as a function of the
actions that were taken in period: 39
T
10
8
# of strategies = 3
history
ㅡ
-
}
~
"
# of strategies = mn
m = # of actions
n = # of decision nodes
45
[ TA
BY
=
,
,
three
decision
⑧
@
@
T
T
T
T
B
B
T
B
B
B
T
T
T
B
B
T
B
B
F
#
sfrafegies
of
3
=
2
=
8
nodes
Game Tree?
K비들들를
:
ㅡ
0
,
46
Nash Equilibrium
}
Two N.E. in the stage game
}
}
}
(M,C), (B, R)
cf. best outcome (T, L) à Dilemma
N.E. in the twice repeated game
}
}
}
}
}
1.
2.
3.
4.
5.
}
}
47
(M, C) in both periods: (M, all M) & (C, all C)
(B, R) in both periods: (B, all B) & (R, all R)
(M, C) & (B, R): (M, all B) & (C, all R)
(B, R) & (M, C): (B, all M) & (R, all C)
(T, L) & (M, C)
Player 1: T in period 1; In period 2, play M if period 1 actions were
(T, L); otherwise play B à (T, M after TL and otherwise B)
Player 2: L in period 1; In period 2, play C if period 1 actions were (T,
L); otherwise play R à (L, C after TL and otherwise R)
Check Optimality
}
Player 1 (against (L, C after TL and otherwise
R))
}
}
}
}
No discounting
If choose M in period 1 à get 6 + 1 = 7
If choose T in period 1 à get 5 + 4 = 9
So, he will choose T in period 1, and choose M in
period 2.
Player 2 (against (T, M after TL and otherwise
B))
}
}
}
48
If choose C in period 1 à get 6 + 1 = 7
If choose L in period 1 à get 5 + 4 = 9
So, he will choose L in period 1, and choose C in
period 2.
Cooperation in Repeated Games
}
}
Because players can react to other players’ past
actions, repeated games allow for equilibrium
outcomes that would not be an equilibrium in
the corresponding one-shot game.
2 x 2 game
}
}
49
Finitely repeated à No cooperation
Infinitely repeated à Cooperation is possible if
players’ discount factor is sufficiently high.
Prisoner’s Dilemma
50
C
D
C
5, 5
3, 6
D
6, 3
4, 4
Twice Repeated
}
2nd period
}
}
Uinque NE: (D, D)
-
oinal
period
1st period
}
}
51
The play in the 1st period cannot affect the players’
behavior in the 2nd period.
So each player chooses the dominant strategy for the
stage game. à (D,D)
Infinitely Repeated
}
Trigger strategy
}
}
C as long as the opponent chooses C
D for specific times once the opponent chooses D
}
}
52
Grim(=cruel) strategy: permanent punishment
Tit-for-tat: One time punishment
Grim Strategy
}
grim(=harsh) strategy
}
}
t=1: Play C in the first period
t>1: Play C if the outcomes in the all previous periods
were (C, C); otherwise, play D.
cixic
.
53
c
,
Check Optimality
다 ziGrim
T,
ㅡ
-
-
}
Follow the grim strategy
5
2
PV = 5 + 5d + 5d + ! =
1- d
8
[
G ~iom
e
peviate
x
}
Deviate to D in the first period and follow the grim
strategy from the second period.
4d
5
1
PV = 6 + 4d + 4d + ! = 6 +
<
iff d >
1- d 1- d
2
2
54
A
Experiment (1984)
by Robert Axelrod
}
Invite to submit computer programs that
specified a strategy for playing a PD repeated a
finite but large number (200) of times
}
The number of repetition was not announced to
applicants.
}
League tournament
Winner: Tit-For-Tat by Anatol Rapoport
}
http://en.wikipedia.org/wiki/Tit_for_tat
}
55
Robert Axelrod (1943-)
Political Sicence & Public Policy @ U. of
Michigan
Evolution of Cooperation
56
Anatol Rapoport (1911-2007)
a Russian-born American Jewish,
mathematical psychologist
57
Properties of Tit For Tat
}
Nice
}
}
Provokable
}
}
}
Don’t be the first to defect.
Retaliate D
Forgiving
Clear (Simple)
}
58
Don’t be too clever.
Pricing Game
K-Mart
TOYS
“R”US
59
Low
High
Low
2000, 2000
4000,0
High
0, 4000
3000,3000
Price Matching – Lowest Price Guarantee
K-Mart
L
Toys
H
R us
M
60
L
H
M
2000,2000
4000,0
2000,2000
0,4000
3000,3000
3000,3000
2000,2000
3000,3000
3000,3000
}
Match:
}
}
Charge H
But match if the rival charge L
Two NE = (L, L) & (M, M)
à Matching strategy is a kind of punishment
mechanism
}
61
}
Match:
}
}
Charge H
But match if the rival charge L
Two NE = (L, L) & (M, M)
à Matching strategy is a kind of punishment
mechanism
}
61